Are there books introducing to Complex Analysis for people with algebraic background?

Question

I'm a third year student who is mostly interested in commutative algebra. In Algebraic Geometry a lot of example come from Complex Analysis. So to deepen my understanding/intuition, I'll finally attend an Introduction to Complex Analysis lecture. (Which is actually made for second year students, but let's say I've not been a big fan of Analysis so far)

Therefore I am looking for a book on the topic of complex analysis which is particulary fit for people who are interested in the more algebraic aspects/ intuition for algebraic geometry and might even have some background in that direction, but still offers a good and complete introduction to complex analysis itself.

I know that a similar topic had already been posted, but I think the lecture I will attend - and therefore the kind of book I am looking for - is even a bit more elementary as it doesn't deal with manifolds at all. (I'm aware of the fact that 'nice connections' with algebraic geometry most likely will only appear when you start studying smooth manifolds, but sadly this is not what this lecture offers).

Answer 1

I don't know how algebraic you can really make a first course in complex analysis. You can certainly start imitating some of the techniques used to study complex manifolds, which have an algebraic flair, but I think this would only obfuscate the underlying analysis--the truly important part.

When complex geometry comes up in algebraic geometry, it's not in purely algebraic terms. Usually the complex geometry comes up precisely because the analysis clarifies/does things for us that the algebra alone was ill-equipped to handle.

In my humblest of opinions, it is a common mistake to rush too quickly into algebraic geometry, complex or otherwise, without having a solid analytic background. These analytic techniques motivate, and inspire much of the modern algebraic theory.

That said, I am an algebraic-y person, and I really enjoyed these notes of Schlag. They are fast paced, and certainly written with an analytic viewpoint, but are supremely rigorous and organized. If you stick to the end, then you'll even have a good understanding of the analytic theory of Riemann surfaces. This is pivotal if you want to understand complex analytic geometry, and it's relation to algebraic geometry (Riemann surfaces are where the strongest connection between algebraic geometry and complex geometry holds).